Turing machine is mathematical model for computation. Deterministic Turing machine (DTM) is computationally equivalent to Non-Deterministic Turing machine (NTM) but the time complexity may not be the same.

>What I'm ultimately interested in is if we, as humans, are operating in a realm that logic cannot operate in. If that makes any sense?

Any mix of logic and randomness is still a machine. You must assume something spiritual if you think there exist rational action without computation behind it. Randomness does not help there.